I found 4 different integers that satisfied the equation:
1) 44+56=100
x=8 y=4
2)-21+121=100
x=-3 y=11
3)-33+133=100
x=19 y=-3
4)-98+198=100
x=-14 y=18
After observing the values of x and y I listed them in order:
Values: x y
-14 18
-3 11
8 4
19 -3
30 -10
41 -17
52 -24
After seeing the values placed in order I noticed a pattern between the values increasing as decreasing, in x, from 8 to 19 it added 11 and from 19 to 30 also. In y, from 4 to -3 it subtracted 7 as in -3 from -10 it subtracted 7. Therefore I could make a rule:
x = 8 + 11(n - 1)
y = 4 - 7(n - 1)
Proving the rule:
n = 4
x = 8 + 11(4-1)
y = 4 - 7(4-1)
x = 8 + 11(3)
y = 4 - 7(3)
x = 8 + 33
y = 4 - 21
x = 41
y = -17
7(41) + 11(-17) = 100
287 + -187 = 100
After testing the rule I could see that it is right and there are infinite integers for 7x + 11y = 100
To investigate if there is a pattern or a rule with the x and y integers.
At first I searched for both positive integers for x and y:
11+89=100
22+78=100
33+67=100
44+56=100
55+45=100
66+34=100
77+23=100
88+12=100
99+1=100
I could only find one postive integer that satisfied the equation;
x: 8
y: 4
After finding only two positive integers I continued my search with positive and negative integers for x and y:
-11+111=100
-21+121=100
-22+122=100
-33+133=100
-44+144=100
-55+155=100
-66+166=100
-77+177=100
-88+188=100
-98+198=100
-99+199=100
I found 4 different integers that satisfied the equation:
1) 44+56=100
x=8 y=4
2)-21+121=100
x=-3 y=11
3)-33+133=100
x=19 y=-3
4)-98+198=100
x=-14 y=18
After observing the values of x and y I listed them in order:
Values:
x y
-14 18
-3 11
8 4
19 -3
30 -10
41 -17
52 -24
After seeing the values placed in order I noticed a pattern between the values increasing as decreasing, in x, from 8 to 19 it added 11 and from 19 to 30 also. In y, from 4 to -3 it subtracted 7 as in -3 from -10 it subtracted 7. Therefore I could make a rule:
x = 8 + 11(n - 1)
y = 4 - 7(n - 1)
Proving the rule:
n = 4
x = 8 + 11(4-1)
y = 4 - 7(4-1)
x = 8 + 11(3)
y = 4 - 7(3)
x = 8 + 33
y = 4 - 21
x = 41
y = -17
7(41) + 11(-17) = 100
287 + -187 = 100
After testing the rule I could see that it is right and there are infinite integers for 7x + 11y = 100